## Picture That!

#### by Arthur Lewbel

Most photographs of juggling show objects scattered more or less randomly in the air, but occasionally something odd happens. In a picture of an ordinary cascade objects may be neatly arranged in pairs in the air, or clubs may be all horizontal or all vertical at the same time. Why does this happen?

Start with a three ball cascade, numbering the balls in the order that they are originally thrown. When ball two is at the top of its arc, ball three is being thrown and ball one is being caught. If a picture is taken when ball three is just as close to one hand (on the way up) as ball one is to the other hand (on the way down), the resulting photo will have a single ball on top and two in a horizontal pair near or in the hands.

A similar effect can happen with any number of balls. Consider a five ball cascade that was started with 3 balls in the right hand and two in the left. When ball three (on its way to the left from the right) is at the top of its arc, ball one is being caught in the left hand and ball five is leaving the right, so balls one and five form a horizontal pair near or in the hands. In between are balls two and four. They will also be in a horizontal row, with ball two heading down to the right and ball four going up from the left. Balls two and four are both the same height because they are both equally distant from ball three in time, and hence are both equally distant from the top of the pattern.

In an N ball (or any object) cascade for any odd N, when one ball is right at the top of its flight, the remaining balls are arranged in (N-1)/2 horizontal pairs. In each pair, one object is going up, the other down, and the pairs themselves alternate so that if one pair has the object on the right going up, then the next pair up has the object on the left going up.

This pairing of objects can be seen in the accompanying photo of Demetrius Alcarese attempting seven clubs. Notice also the vertical distances between these pairs of clubs. The higher up one looks, the closer is one pair of clubs to the next pair up (or to the single club on top). This occurs because adjacent throws are equally spaced from each other in time. When an object is thrown upward it slows down, stops at the top of its flight, then speeds up on the way down until it is caught. So, to be equally spaced in time the objects need to be closer together vertically near the top of the flight than near the bottom.

This explains pairing, but what about all clubs being vertical at the same time, like in the photo of Larry Merlo? A mathematically detailed explanation can be found in "The Physics of Juggling," by Bengt Magnusson and Bruce Tiemann (The Physics Teacher, November, 1989), but the basic idea is simple.

Consider a three club cascade with single spins. Divide the time between when a single club is thrown once and when it is thrown again into three periods (call the time periods A, B, and C). In period A the club is in the hand. In period B the club makes its first half turn in the air, and in period C it makes its second half turn. Since this is a three club cascade with clubs equally spaced in time, when one club is doing A, another is doing B and the third is doing C. If you take a photo right in the middle of a time period, the club doing B will be vertical with its knob down (being in the middle of its first half turn) while the club doing C will be vertical with its knob up (the middle of its second half turn), so both of the clubs in the air will be vertical at the same time.

The same thing can happen with 5 club double spins (now dividing into five periods) or with 7 club triple spins. For example, with five clubs, between the time club one is thrown and caught it makes four half spins, and in that same time period exactly four other clubs are also being thrown and caught, making each club exactly a half spin ahead of the previous one.

Given how simple a cascade juggling photo can look, do you suppose we could ever get casual artists to draw some juggling pattern other than a shower?

"The Academic Juggler" is an occasional feature of Juggler's World, and is devoted to all kinds of formal analyses of juggling. Anybody who has suggestions, comments, or potential contributions for this feature is encouraged to write to Arthur Lewbel; 3 Audubon Rd.; Lexington MA 02173; 617/862-3089; or E-mail (Bitnet address Lewbel@Brandeis). Please include your own phone number if you write.

Academic Juggler / Index, Vol. 43, No. 3 / jis@juggling.org